Spectra of weighted composition operators with automorphic symbols Mikael Lindstr¨ om Department of Mathematical Sciences University of Oulu Joint work with Olli Hyv¨ arinen, Ilmari Nieminen and Erno Saukko Gothenburg, July/August, 2013
We denote by H ( D ) the space of analytic functions on the open unit disc D of the complex plane. For ϕ an analytic selfmap of D and u in H ( D ) , the weighted composition operator uC ϕ is defined by ( uC ϕ ) f ( z ) = u ( z ) f ( ϕ ( z )) , where f ∈ H ( D ) . There are two particularly interesting special cases of such operators: on one hand, taking u = 1 gives the composition operator C ϕ , and on the other, putting ϕ = id , the identity function on D , gives us the multiplication operator M u . 2
The study of composition operators on analytic function spaces was started about 40 years ago by Nordgren, Kamowitz, Shapiro, Cowen and MacCluer and others. There are at least two main goals by studying weighted composition operators. • Relate properties of the functions u, ϕ to properties of the operators C ϕ and uC ϕ . • Relate properties of the operators C ϕ and uC ϕ to properties of other operators, in order to better understand other operators. 3
For invertible uC ϕ , we aim to • calculate the spectral radius of uC ϕ ; and • determine its spectrum . The methods of proof should be general, so that the results hold on many large spaces of analytic functions. In fact, we have designed a unified approach to determine the spectra of invertible weighted composition operators on a broad class A of analytic function spaces. The spectrum of uC ϕ was studied by Gunatillake (2011) on the Hardy-Hilbert space H 2 ( D ) and by Kamowitz (1978) on the disc algebra A ( D ) . 4
Spectral properties of uC ϕ depend on the type of the symbol ϕ . These types are: • elliptic - ϕ has fixed point in D ; • parabolic - ϕ has a unique fixed point in ∂ D ; • hyperbolic - ϕ has two fixed point in ∂ D . The hyperbolic case is the most interesting one. Fact (1987, Nordgren, Rosenthal, Wintrobe) : every linear bounded operator T has a closed non-trivial invariant subspace ⇔ the minimal non-trivial closed invariant subspaces for C ϕ on H 2 ( D ) are one-dimensional, where ϕ is a hyperbolic automorphism of D . 5
Let A ⊂ H ( D ) be a Banach space; its norm is denoted by || · || A . Suppose there is a constant s > 0 such that the following hold: 1) For each f ∈ A and z ∈ D we have | f ( z ) | � || f || A (1 − | z | 2 ) − s . 2) For each z ∈ D there is some f z ∈ A with || f z || A ≤ 1 such that f z ( z )(1 − | z | 2 ) s = 1 . 3) For ϕ an automorphism, || C ϕ || � (1 − | ϕ (0) | 2 ) − s . 4) For w ∈ H ∞ ( D ) we have || M w || ≤ || w || ∞ . 6
The following spaces satisfy all the stated conditions. • Hardy spaces H p ( D ) , 1 ≤ p < ∞ , with s = 1 p . • Weighted Bergman spaces A p α ( D ) , p ≥ 1 , α > − 1 , consisting of all analytic functions f on D such that � | f ( z ) | p (1 − | z | ) α dA ( z ) < ∞ . D Here s = ( α + 2) /p. • The weighted Banach spaces of analytic functions H ∞ p ( D ) , 0 < p < ∞ , consisting of analytic functions f on D with | f ( z ) | (1 − | z | ) p < ∞ . sup z ∈ D Here s = p. 7
Theorem 1. For such A and ϕ an analytic selfmap of D , the weighted composition operator uC ϕ is Fredholm on A if and only if M u is Fredholm and ϕ is an automorphism of the unit disc. In the rest of the talk we will assume that uC ϕ : A → A is invertible and therefore ϕ is an automorphism. We have: Theorem 2. For such A and an automorphism ϕ the operator uC ϕ : A → A is invertible if and only if u is bounded and bounded away from zero. The inverse is given by 1 ( uC ϕ ) − 1 = u ◦ ϕ − 1 C ϕ − 1 . 8
Concerning the boundary fixed points of ϕ , one result of particular importance is the celebrated Denjoy-Wolff theorem. If ϕ is either a parabolic or a hyperbolic automorphism of D , this theorem guarantees that there is a (unique) fixed point a ∈ ∂ D such that n →∞ ϕ n ( z ) = a lim uniformly on compact subsets of D ; the point a is called the Denjoy-Wolff point of ϕ . Moreover, • if ϕ is parabolic, then ϕ ′ ( a ) = 1 ; and • if ϕ is hyperbolic and its other fixed point b ∈ ∂ D , then 0 < ϕ ′ ( a ) < 1 and ϕ ′ ( a ) = 1 /ϕ ′ ( b ) . 9
The parabolic case Theorem 3. Let A be any of the spaces A p α ( D ) , p ≥ 1 , α > − 1 , and s = α +2 p ; H p ( D ) , p ≥ 1 , and s = 1 p ; or H ∞ p ( D ) , 0 < p < ∞ , and s = p . Suppose that u ∈ A ( D ) is bounded away from zero on D and let ϕ be a parabolic automorphism of D whose Denjoy-Wolff point is a ∈ ∂ D . Then the spectral radius r ( uC ϕ ) = | u ( a ) | and the spectrum σ A ( uC ϕ ) is the circle σ A ( uC ϕ ) = { λ ∈ C ; | λ | = | u ( a ) |} . 10
The hyperbolic case Theorem 4. Let A be any of the spaces A p α ( D ) , p ≥ 1 , α > − 1 , and s = α +2 p ; H p ( D ) , p ≥ 1 , and s = 1 p ; or H ∞ p ( D ) , 0 < p < ∞ , and s = p . Let ϕ be a hyperbolic automorphism of D with fixed points a (attractive) and b (repulsive) in ∂ D . If u ∈ A ( D ) is bounded away from zero, then � � ϕ ′ ( b ) s , | u ( a ) | | u ( b ) | r ( uC ϕ ) = max ϕ ′ ( a ) s and if, moreover, | u ( b ) /ϕ ′ ( b ) s | ≤ | u ( a ) /ϕ ′ ( a ) s | , then � � λ ∈ C ; | u ( b ) | ϕ ′ ( b ) s ≤ | λ | ≤ | u ( a ) | σ A ( uC ϕ ) = . ϕ ′ ( a ) s 11
The elliptic case Theorem 5. Let A be any of the spaces A p α ( D ) , p ≥ 1 , α > − 1 , and s = α +2 p ; H p ( D ) , p ≥ 1 , and s = 1 p ; or H ∞ p ( D ) , 0 < p < ∞ , and s = p . ( a ) Suppose that u ∈ A ( D ) and ϕ is an automorphism of D such that there is a positive integer j with ϕ j ( z ) = z for all z ∈ D . If n is the smallest such integer, then σ A ( uC ϕ ) = { λ ∈ C ; λ n = u ( n ) ( z ) , z ∈ D } , where u ( n ) = � n − 1 m =0 u ◦ ϕ m ∈ H ( D ) . ( b ) Suppose u ∈ A ( D ) is bounded away from zero on D and let ϕ be an elliptic automorphism such that ϕ n ( z ) �≡ z for all positive integers n . If a ∈ D is the unique fixed point of ϕ , then σ A ( uC ϕ ) = { λ ∈ C ; | λ | = | u ( a ) |} . 12
Let us now consider two examples in which we determine the spectrum of a weighted composition operator uC ϕ . (1) Let uC ϕ be a unitary weighted composition operator on H 2 ( D ) , where ϕ is an automorphism. Then s = 1 / 2 . Since uC ϕ is unitary, it can be shown that � 1 − | z 0 | 2 u ( z ) = c , 1 − z 0 z where ϕ ( z 0 ) = 0 and | c | = 1 . Moreover, r ( uC ϕ ) = 1 and σ ( uC ϕ ) ⊆ ∂ D . ( a ) Suppose that ϕ ( z ) = µz , where | µ | = 1 . Then u ( z ) = c . If µ j = 1 for some positive integer j , then if n is the smallest such integer, the previous Theorem 5 gives σ H 2 ( D ) ( uC ϕ ) = { λ ; λ n = c n } = { µ k c ; k = 0 , 1 , . . . , n − 1 } . If there is no such integer, we get σ H 2 ( D ) ( uC ϕ ) = { λ ; | λ | = 1 } . 13
( b ) Let ϕ ( z ) = (1+ i ) z − 1 z + i − 1 , that is, ϕ is a parabolic automorphism of D . 1 Then z 0 = 1+ i . By Theorem 3 r ( uC ϕ ) = | u (1) | = 1 and the spectrum σ H 2 ( D ) ( uC ϕ ) = { λ ; | λ | = 1 } . ( c ) Consider the hyperbolic automorphism ϕ ( z ) = z + r 1+ rz where 0 < r < 1 , that is, ϕ is a hyperbolic automorphism of D with Denjoy-Wolff point a = 1 , and the other fixed point b = − 1 . Then z 0 = − r . In this case | u ( − 1) | > | u (1) | , and ϕ ′ (1) 1 / 2 = | u ( − 1) | | u (1) | ϕ ′ ( − 1) 1 / 2 = 1 = r ( uC ϕ ) . Theorem 4 gives us that the spectrum of uC ϕ is the unit circle. 14
(2) Let us consider the space H ∞ p ( D ) , which is non-Hilbert. Now s = p . Let ϕ ( z ) = z + r 1+ rz with 0 < r < 1 , so again ϕ is a hyperbolic automorphism of D with fixed points a = 1 and b = − 1 . Put u ( z ) = ( ϕ ′ ( z )) p = (1 − r 2 ) p / (1 + rz ) 2 p . Then u ∈ A ( D ) is bounded away from zero, and so uC ϕ is invertible on H ∞ p ( D ) . By Theorem 3, the spectral radius of uC ϕ is � � ϕ ′ ( − 1) p , | u (1) | | u ( − 1) | r ( uC ϕ ) = max = 1 ϕ ′ (1) p and σ H ∞ p ( uC ϕ ) = { λ ∈ C ; | λ | = 1 } . Finally we briefly discuss the proof of � � ϕ ′ ( a ) s , | u ( b ) | | u ( a ) | r ( uC ϕ ) = max , ϕ ′ ( b ) s with hyperbolic automorphism ϕ ( z ) = z + r 1+ rz , 0 < r < 1 . 15
One only need to show that r ( uC ϕ ) ≤ max � | u ( a ) | ϕ ′ ( a ) s , | u ( b ) | � . We ϕ ′ ( b ) s proceed as follows. For any n ∈ N we have n − 1 n − 1 n − 1 u ◦ ϕ j � � � || ( uC ϕ ) n || = || ( ϕ ′ ◦ ϕ j ) s C ϕ n || . u ◦ ϕ j C ϕ n || = || ( ϕ ′ ◦ ϕ j ) s · j =0 j =0 j =0 j =0 ϕ ′ ◦ ϕ j = ( ϕ n ) ′ . Further, w =: Notice that � n − 1 u ( ϕ ′ ) s and observe that w ∈ A ( D ) is also bounded away from zero. By condition 4) , we obtain || ( uC ϕ ) n || 1 /n ≤ || w ( n ) || 1 /n ∞ || ( ϕ ′ n ) s C ϕ n || 1 /n . Since it can be shown that � � � � ϕ ′ ( a ) s , | u ( b ) | | u ( a ) | n →∞ || w ( n ) || 1 /n lim ∞ = max | w ( a ) | , | w ( b ) | = max ϕ ′ ( b ) s n ) s C ϕ n || 1 /n ≤ 1 , the claim follows. and lim n →∞ || ( ϕ ′ 16
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